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Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$

Jonas Haferkamp

TL;DR

The paper shows that random quantum circuits on $n$ qubits form approximate unitary $t$-designs in depth $O(n t^{5+o(1)})$, improving the previous $O(n t^{10.5})$ bound by refining spectral-gap analyses. The authors introduce an unconditional gap bound and a tight reduction using an auxiliary walk that alternates Clifford operations with Haar randomness, employing path coupling and the Clifford group’s properties alongside Gao’s quantum union bound. These techniques yield explicit depth bounds for local and brickwork architectures and translate into strong lower bounds on quantum circuit complexity, $C_{\delta}(U)$, scaling as $\Omega(nt)$, and sublinear-depth growth results for complexity up to exponential circuit depths. The work also outlines open questions about unconditional gaps and the role of Clifford representations, suggesting avenues for further tightening of the design-depth scaling. Overall, the results deepen the connection between spectral-gap properties, unitary designs, and circuit complexity, with potential implications for randomness generation and information scrambling in quantum systems.

Abstract

The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary $t$-designs. Unitary $t$-designs are probability distributions that mimic Haar randomness up to $t$th moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth $O(n t^{10.5})$ are approximate unitary $t$-designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by $Ω(n^{-1}t^{-9.5})$. We improve this lower bound to $Ω(n^{-1}t^{-4-o(1)})$, where the $o(1)$ term goes to $0$ as $t\to\infty$. A direct consequence of this scaling is that random quantum circuits generate approximate unitary $t$-designs in depth $O(nt^{5+o(1)})$. Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.

Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$

TL;DR

The paper shows that random quantum circuits on qubits form approximate unitary -designs in depth , improving the previous bound by refining spectral-gap analyses. The authors introduce an unconditional gap bound and a tight reduction using an auxiliary walk that alternates Clifford operations with Haar randomness, employing path coupling and the Clifford group’s properties alongside Gao’s quantum union bound. These techniques yield explicit depth bounds for local and brickwork architectures and translate into strong lower bounds on quantum circuit complexity, , scaling as , and sublinear-depth growth results for complexity up to exponential circuit depths. The work also outlines open questions about unconditional gaps and the role of Clifford representations, suggesting avenues for further tightening of the design-depth scaling. Overall, the results deepen the connection between spectral-gap properties, unitary designs, and circuit complexity, with potential implications for randomness generation and information scrambling in quantum systems.

Abstract

The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary -designs. Unitary -designs are probability distributions that mimic Haar randomness up to th moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth are approximate unitary -designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by . We improve this lower bound to , where the term goes to as . A direct consequence of this scaling is that random quantum circuits generate approximate unitary -designs in depth . Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.
Paper Structure (9 sections, 11 theorems, 61 equations, 1 figure)

This paper contains 9 sections, 11 theorems, 61 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Revisiting the spectral gap of random quantum circuits
  3. Results
  4. Tensor product expanders and unitary designs
  5. Quantum circuit complexity
  6. Proof of Theorem \ref{['thm:finalgapbounds']} and Theorem \ref{['thm:unconditionalgap']}
  7. Outlook: Gaps from representations of the Clifford group?
  8. Acknowledgements
  9. Proof of Observation \ref{['observation:improvedreduction']}

Key Result

Lemma 1

Let $\nu$ be a probability distribution on $U(d)$ such that $g(\nu,t)\leq \varepsilon/d^{2t}$. Then $\nu$ is an $\varepsilon$-approximate unitary $t$-design and obeys both Eq. eq:approxdesign1 and Eq. eq:approxdesign2.

Figures (1)

  • Figure 1: Four steps of the random walk $\sigma$ on $n=6$ qubits.

Theorems & Definitions (20)

  • Definition 1: Approximate unitary designs
  • Lemma 1
  • Definition 2: Random quantum circuits
  • Lemma 2: Detectability lemma and union bound
  • Theorem 1: Unconditional gap
  • Theorem 2: Tensor product expanders
  • proof
  • Corollary 1: Unitary designs
  • Definition 3: Quantum circuit complexity
  • Theorem 3: Informal, Ref. brandao_local_2016
  • ...and 10 more